cp's OEIS Frontend

A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

%I A383309 #7 Apr 25 2025 20:08:55
%S A383309 1,3,5,7,9,11,17,19,23,25,27,31,35,41,49,53,59,67,81,83,97,103,109,
%T A383309 121,125,127,131,157,175,179,191,209,211,227,241,243,245,277,283,289,
%U A383309 311,331,343,353,361,367,391,401,419,431,461,509,529,547,563,587,599
%N A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.
%C A383309 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%F A383309 Equals A326534 /\ A355743.
%e A383309 The systems with these MM-numbers begin:
%e A383309    1: {}
%e A383309    3: {{1}}
%e A383309    5: {{2}}
%e A383309    7: {{1,1}}
%e A383309    9: {{1},{1}}
%e A383309   11: {{3}}
%e A383309   17: {{4}}
%e A383309   19: {{1,1,1}}
%e A383309   23: {{2,2}}
%e A383309   25: {{2},{2}}
%e A383309   27: {{1},{1},{1}}
%e A383309   31: {{5}}
%e A383309   35: {{2},{1,1}}
%e A383309   41: {{6}}
%e A383309   49: {{1,1},{1,1}}
%e A383309   53: {{1,1,1,1}}
%e A383309   59: {{7}}
%e A383309   67: {{8}}
%e A383309   81: {{1},{1},{1},{1}}
%e A383309   83: {{9}}
%e A383309   97: {{3,3}}
%t A383309 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383309 Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@PrimePowerQ/@prix[#]&]
%Y A383309 Twice-partitions of this type are counted by A279789.
%Y A383309 For just a common sum we have A326534.
%Y A383309 For just constant blocks we have A355743.
%Y A383309 Numbers without a factorization of this type are listed by A381871, counted by A381993.
%Y A383309 The multiplicative version is A381995.
%Y A383309 This is the odd case of A382215.
%Y A383309 For strict instead of constant blocks we have A382304.
%Y A383309 A001055 counts factorizations, strict A045778.
%Y A383309 A023894 counts partitions into prime-powers.
%Y A383309 A034699 gives maximal prime-power divisor.
%Y A383309 A050361 counts factorizations into distinct prime powers.
%Y A383309 A055396 gives least prime index, greatest A061395.
%Y A383309 A056239 adds up prime indices, row sums of A112798.
%Y A383309 A246655 lists the prime-powers (A000961 includes 1), towers A164336.
%Y A383309 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A383309 A353864 counts rucksack partitions, ranked by A353866.
%Y A383309 A355742 chooses a prime-power divisor of each prime index.
%Y A383309 Cf. A000688, A000720, A001222, A006171, A038041, A279784, A302242, A302493, A321455, A326518, A381719.
%K A383309 nonn
%O A383309 1,2
%A A383309 _Gus Wiseman_, Apr 25 2025